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Relative Velocity

Find the area...

Question

Find the area bounded by $x -$ axis and an arc of the cycloid
$x = a (2 t - sin 2 t), y = a (1 - cos 2 t)$

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Solution

for cycloid curve

Using Green's theorem

$= \int C_{1} - y d x$

$= \int_{2 π}^{0} (a (1 - c o s t) \times a (1 - c o s t)) d t$

$= a^{2} \int_{2 π}^{0} 1 - 2 c o t t - c o s^{2} t d t$

by solving
$= 3 π a^{2}$

$\Rightarrow \int C_{2} - y d x$

$\Rightarrow - \int_{2}^{0} 0 \times a (1 - c o s t) d t = 0$

$a r e a = 3 π a^{2} + 0 = 3 π a^{2}$

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